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Appropriate Combination of Crossover Operator and Mutation Operator in Genetic Algorithms for the Travelling Salesman Problem
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作者 Zakir Hussain Ahmed Habibollah Haron Abdullah Al-Tameem 《Computers, Materials & Continua》 SCIE EI 2024年第5期2399-2425,共27页
Genetic algorithms(GAs)are very good metaheuristic algorithms that are suitable for solving NP-hard combinatorial optimization problems.AsimpleGAbeginswith a set of solutions represented by a population of chromosomes... Genetic algorithms(GAs)are very good metaheuristic algorithms that are suitable for solving NP-hard combinatorial optimization problems.AsimpleGAbeginswith a set of solutions represented by a population of chromosomes and then uses the idea of survival of the fittest in the selection process to select some fitter chromosomes.It uses a crossover operator to create better offspring chromosomes and thus,converges the population.Also,it uses a mutation operator to explore the unexplored areas by the crossover operator,and thus,diversifies the GA search space.A combination of crossover and mutation operators makes the GA search strong enough to reach the optimal solution.However,appropriate selection and combination of crossover operator and mutation operator can lead to a very good GA for solving an optimization problem.In this present paper,we aim to study the benchmark traveling salesman problem(TSP).We developed several genetic algorithms using seven crossover operators and six mutation operators for the TSP and then compared them to some benchmark TSPLIB instances.The experimental studies show the effectiveness of the combination of a comprehensive sequential constructive crossover operator and insertion mutation operator for the problem.The GA using the comprehensive sequential constructive crossover with insertion mutation could find average solutions whose average percentage of excesses from the best-known solutions are between 0.22 and 14.94 for our experimented problem instances. 展开更多
关键词 Travelling salesman problem genetic algorithms crossover operator mutation operator comprehensive sequential constructive crossover insertion mutation
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A Novel Insertion Solution for the Travelling Salesman Problem
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作者 Emmanuel Oluwatobi Asani Aderemi Elisha Okeyinka +5 位作者 Sunday Adeola Ajagbe Ayodele Ariyo Adebiyi Roseline Oluwaseun Ogundokun Temitope Samson Adekunle Pragasen Mudali Matthew Olusegun Adigun 《Computers, Materials & Continua》 SCIE EI 2024年第4期1581-1597,共17页
The studypresents theHalfMax InsertionHeuristic (HMIH) as a novel approach to solving theTravelling SalesmanProblem (TSP). The goal is to outperform existing techniques such as the Farthest Insertion Heuristic (FIH) a... The studypresents theHalfMax InsertionHeuristic (HMIH) as a novel approach to solving theTravelling SalesmanProblem (TSP). The goal is to outperform existing techniques such as the Farthest Insertion Heuristic (FIH) andNearest Neighbour Heuristic (NNH). The paper discusses the limitations of current construction tour heuristics,focusing particularly on the significant margin of error in FIH. It then proposes HMIH as an alternative thatminimizes the increase in tour distance and includes more nodes. HMIH improves tour quality by starting withan initial tour consisting of a ‘minimum’ polygon and iteratively adding nodes using our novel Half Max routine.The paper thoroughly examines and compares HMIH with FIH and NNH via rigorous testing on standard TSPbenchmarks. The results indicate that HMIH consistently delivers superior performance, particularly with respectto tour cost and computational efficiency. HMIH’s tours were sometimes 16% shorter than those generated by FIHand NNH, showcasing its potential and value as a novel benchmark for TSP solutions. The study used statisticalmethods, including Friedman’s Non-parametric Test, to validate the performance of HMIH over FIH and NNH.This guarantees that the identified advantages are statistically significant and consistent in various situations. Thiscomprehensive analysis emphasizes the reliability and efficiency of the heuristic, making a compelling case for itsuse in solving TSP issues. The research shows that, in general, HMIH fared better than FIH in all cases studied,except for a few instances (pr439, eil51, and eil101) where FIH either performed equally or slightly better thanHMIH. HMIH’s efficiency is shown by its improvements in error percentage (δ) and goodness values (g) comparedto FIH and NNH. In the att48 instance, HMIH had an error rate of 6.3%, whereas FIH had 14.6% and NNH had20.9%, indicating that HMIH was closer to the optimal solution. HMIH consistently showed superior performanceacross many benchmarks, with lower percentage error and higher goodness values, suggesting a closer match tothe optimal tour costs. This study substantially contributes to combinatorial optimization by enhancing currentinsertion algorithms and presenting a more efficient solution for the Travelling Salesman Problem. It also createsnew possibilities for progress in heuristic design and optimization methodologies. 展开更多
关键词 Nearest neighbour heuristic farthest insertion heuristic half max insertion heuristic tour construction travelling salesman problem
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Solving the Generalized Traveling Salesman Problem Using Sequential Constructive Crossover Operator in Genetic Algorithm
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作者 Zakir Hussain Ahmed Maha Ata Al-Furhood +1 位作者 Abdul Khader Jilani Saudagar Shakir Khan 《Computer Systems Science & Engineering》 2024年第5期1113-1131,共19页
The generalized travelling salesman problem(GTSP),a generalization of the well-known travelling salesman problem(TSP),is considered for our study.Since the GTSP is NP-hard and very complex,finding exact solutions is h... The generalized travelling salesman problem(GTSP),a generalization of the well-known travelling salesman problem(TSP),is considered for our study.Since the GTSP is NP-hard and very complex,finding exact solutions is highly expensive,we will develop genetic algorithms(GAs)to obtain heuristic solutions to the problem.In GAs,as the crossover is a very important process,the crossovermethods proposed for the traditional TSP could be adapted for the GTSP.The sequential constructive crossover(SCX)and three other operators are adapted to use in GAs to solve the GTSP.The effectiveness of GA using SCX is verified on some GTSP Library(GTSPLIB)instances first and then compared against GAs using the other crossover methods.The computational results show the success of the GA using SCX for this problem.Our proposed GA using SCX,and swap mutation could find average solutions whose average percentage of excesses fromthe best-known solutions is between 0.00 and 14.07 for our investigated instances. 展开更多
关键词 Generalized travelling salesman problem NP-HARD genetic algorithms sequential constructive crossover swap mutation
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On the “Onion Husk” Algorithm for Approximate Solution of the Traveling Salesman Problem
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作者 Mikhail E. Abramyan Nikolai I. Krainiukov Boris F. Melnikov 《Journal of Applied Mathematics and Physics》 2024年第4期1557-1570,共14页
The paper describes some implementation aspects of an algorithm for approximate solution of the traveling salesman problem based on the construction of convex closed contours on the initial set of points (“cities”) ... The paper describes some implementation aspects of an algorithm for approximate solution of the traveling salesman problem based on the construction of convex closed contours on the initial set of points (“cities”) and their subsequent combination into a closed path (the so-called contour algorithm or “onion husk” algorithm). A number of heuristics related to the different stages of the algorithm are considered, and various variants of the algorithm based on these heuristics are analyzed. Sets of randomly generated points of different sizes (from 4 to 90 and from 500 to 10,000) were used to test the algorithms. The numerical results obtained are compared with the results of two well-known combinatorial optimization algorithms, namely the algorithm based on the branch and bound method and the simulated annealing algorithm. . 展开更多
关键词 Branch and Bound Method Contour Algorithm “Onion Husk” Algorithm Simulated Annealing Method Traveling salesman Problem
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K-DSA for the multiple traveling salesman problem 被引量:1
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作者 TONG Sheng QU Hong XUE Junjie 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2023年第6期1614-1625,共12页
Aimed at a multiple traveling salesman problem(MTSP)with multiple depots and closed paths,this paper proposes a k-means clustering donkey and a smuggler algorithm(KDSA).The algorithm first uses the k-means clustering ... Aimed at a multiple traveling salesman problem(MTSP)with multiple depots and closed paths,this paper proposes a k-means clustering donkey and a smuggler algorithm(KDSA).The algorithm first uses the k-means clustering method to divide all cities into several categories based on the center of various samples;the large-scale MTSP is divided into multiple separate traveling salesman problems(TSPs),and the TSP is solved through the DSA.The proposed algorithm adopts a solution strategy of clustering first and then carrying out,which can not only greatly reduce the search space of the algorithm but also make the search space more fully explored so that the optimal solution of the problem can be more quickly obtained.The experimental results from solving several test cases in the TSPLIB database show that compared with other related intelligent algorithms,the K-DSA has good solving performance and computational efficiency in MTSPs of different scales,especially with large-scale MTSP and when the convergence speed is faster;thus,the advantages of this algorithm are more obvious compared to other algorithms. 展开更多
关键词 k-means clustering donkey and smuggler algorithm(DSA) multiple traveling salesman problem(MTSP) multiple depots and closed paths.
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An Improved Farmland Fertility Algorithm with Hyper-Heuristic Approach for Solving Travelling Salesman Problem
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作者 Farhad Soleimanian Gharehchopogh Benyamin Abdollahzadeh Bahman Arasteh 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第6期1981-2006,共26页
Travelling Salesman Problem(TSP)is a discrete hybrid optimization problem considered NP-hard.TSP aims to discover the shortest Hamilton route that visits each city precisely once and then returns to the starting point... Travelling Salesman Problem(TSP)is a discrete hybrid optimization problem considered NP-hard.TSP aims to discover the shortest Hamilton route that visits each city precisely once and then returns to the starting point,making it the shortest route feasible.This paper employed a Farmland Fertility Algorithm(FFA)inspired by agricultural land fertility and a hyper-heuristic technique based on the Modified Choice Function(MCF).The neighborhood search operator can use this strategy to automatically select the best heuristic method formaking the best decision.Lin-Kernighan(LK)local search has been incorporated to increase the efficiency and performance of this suggested approach.71 TSPLIB datasets have been compared with different algorithms to prove the proposed algorithm’s performance and efficiency.Simulation results indicated that the proposed algorithm outperforms comparable methods of average mean computation time,average percentage deviation(PDav),and tour length. 展开更多
关键词 Travelling salesman problem optimization farmland fertility optimization algorithm Lin-Kernighan
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A Scheme Library-Based Ant Colony Optimization with 2-Opt Local Search for Dynamic Traveling Salesman Problem
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作者 Chuan Wang Ruoyu Zhu +4 位作者 Yi Jiang Weili Liu Sang-Woon Jeon Lin Sun Hua Wang 《Computer Modeling in Engineering & Sciences》 SCIE EI 2023年第5期1209-1228,共20页
The dynamic traveling salesman problem(DTSP)is significant in logistics distribution in real-world applications in smart cities,but it is uncertain and difficult to solve.This paper proposes a scheme library-based ant... The dynamic traveling salesman problem(DTSP)is significant in logistics distribution in real-world applications in smart cities,but it is uncertain and difficult to solve.This paper proposes a scheme library-based ant colony optimization(ACO)with a two-optimization(2-opt)strategy to solve the DTSP efficiently.The work is novel and contributes to three aspects:problemmodel,optimization framework,and algorithmdesign.Firstly,in the problem model,traditional DTSP models often consider the change of travel distance between two nodes over time,while this paper focuses on a special DTSP model in that the node locations change dynamically over time.Secondly,in the optimization framework,the ACO algorithm is carried out in an offline optimization and online application framework to efficiently reuse the historical information to help fast respond to the dynamic environment.The framework of offline optimization and online application is proposed due to the fact that the environmental change inDTSPis caused by the change of node location,and therefore the newenvironment is somehowsimilar to certain previous environments.This way,in the offline optimization,the solutions for possible environmental changes are optimized in advance,and are stored in a mode scheme library.In the online application,when an environmental change is detected,the candidate solutions stored in the mode scheme library are reused via ACO to improve search efficiency and reduce computational complexity.Thirdly,in the algorithm design,the ACO cooperates with the 2-opt strategy to enhance search efficiency.To evaluate the performance of ACO with 2-opt,we design two challenging DTSP cases with up to 200 and 1379 nodes and compare them with other ACO and genetic algorithms.The experimental results show that ACO with 2-opt can solve the DTSPs effectively. 展开更多
关键词 Dynamic traveling salesman problem(DTSP) offline optimization and online application ant colony optimization(ACO) two-optimization(2-opt)strategy
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Use the Power of a Genetic Algorithm to Maximize and Minimize Cases to Solve Capacity Supplying Optimization and Travelling Salesman in Nested Problems
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作者 Ali Abdulhafidh Ibrahim Hajar Araz Qader Nour Ai-Huda Akram Latif 《Journal of Computer and Communications》 2023年第3期24-31,共8页
Using Genetic Algorithms (GAs) is a powerful tool to get solution to large scale design optimization problems. This paper used GA to solve complicated design optimization problems in two different applications. The ai... Using Genetic Algorithms (GAs) is a powerful tool to get solution to large scale design optimization problems. This paper used GA to solve complicated design optimization problems in two different applications. The aims are to implement the genetic algorithm to solve these two different (nested) problems, and to get the best or optimization solutions. 展开更多
关键词 Genetic Algorithm Capacity Supplying Optimization Traveling salesman Problem Nested Problems
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A Comparison of Death of a Salesman & The Glass Menagerie ——The use of expressionistic devices
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作者 李侠 吕娥娥 《陕西师范大学学报(哲学社会科学版)》 CSSCI 北大核心 2004年第S2期409-411,共3页
Expressionism was an aesthetic movement originated in painting in the early 1920’s. But it has also exerted great influence on other forms of arts, including dramatic works. This paper intends to compare the applicat... Expressionism was an aesthetic movement originated in painting in the early 1920’s. But it has also exerted great influence on other forms of arts, including dramatic works. This paper intends to compare the application of expressionistic devices such as flashbacks, and the use of symbols and music in Death of a Salesman and The Glass Menagerie. 展开更多
关键词 EXPRESSIONISM DEATH of a salesman The GLASS Menagerie
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A Free Exchange of Space and Time-An Analysis of Some Creative Techniques in “Death of a Salesman”
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作者 朱晓静 夏春红 《海外英语》 2013年第11X期233-234,共2页
"Death of a Salesman"is important not only in its contents, but also in its creative techniques. This paper will focus on Miller's specific methods with which he depicts Willy's inner world. To be ex... "Death of a Salesman"is important not only in its contents, but also in its creative techniques. This paper will focus on Miller's specific methods with which he depicts Willy's inner world. To be exact, he borrowed some techniques from the films to achieve a free exchange of space and time, that is, to make Willy's remembrance and illusion audio-visual, making the play characteristic of the films. 展开更多
关键词 " DEATH of a salesman" FLASHBACK dissolve MEDIUM
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On the Symbolism in Arthur Miller’s Death of a Salesman
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作者 魏文萍 《海外英语》 2019年第19期182-183,共2页
Death of a Salesman is the greatest masterpiece of Arthur Miller.The play is about a common man trying to reach his American dream and the disillusionment of his dream.This paper mainly analyzes the symbols that are u... Death of a Salesman is the greatest masterpiece of Arthur Miller.The play is about a common man trying to reach his American dream and the disillusionment of his dream.This paper mainly analyzes the symbols that are used in the play to embody the theme. 展开更多
关键词 DEATH of a salesman SYMBOLS AMERICAN DREAM
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Improved ant colony optimization algorithm for the traveling salesman problems 被引量:22
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作者 Rongwei Gan Qingshun Guo +1 位作者 Huiyou Chang Yang Yi 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2010年第2期329-333,共5页
Ant colony optimization (ACO) is a new heuristic algo- rithm which has been proven a successful technique and applied to a number of combinatorial optimization problems. The traveling salesman problem (TSP) is amo... Ant colony optimization (ACO) is a new heuristic algo- rithm which has been proven a successful technique and applied to a number of combinatorial optimization problems. The traveling salesman problem (TSP) is among the most important combinato- rial problems. An ACO algorithm based on scout characteristic is proposed for solving the stagnation behavior and premature con- vergence problem of the basic ACO algorithm on TSP. The main idea is to partition artificial ants into two groups: scout ants and common ants. The common ants work according to the search manner of basic ant colony algorithm, but scout ants have some differences from common ants, they calculate each route's muta- tion probability of the current optimal solution using path evaluation model and search around the optimal solution according to the mutation probability. Simulation on TSP shows that the improved algorithm has high efficiency and robustness. 展开更多
关键词 ant colony optimization heuristic algorithm scout ants path evaluation model traveling salesman problem.
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A Hybrid Genetic Algorithm for the Traveling Salesman Problem with Pickup and Delivery 被引量:10
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作者 Fang-Geng Zhao Jiang-Sheng Sun +1 位作者 Su-Jian Li Wei-Min Liu 《International Journal of Automation and computing》 EI 2009年第1期97-102,共6页
In this paper, a hybrid genetic algorithm (GA) is proposed for the traveling salesman problem (TSP) with pickup and delivery (TSPPD). In our algorithm, a novel pheromone-based crossover operator is advanced that... In this paper, a hybrid genetic algorithm (GA) is proposed for the traveling salesman problem (TSP) with pickup and delivery (TSPPD). In our algorithm, a novel pheromone-based crossover operator is advanced that utilizes both local and global information to construct offspring. In addition, a local search procedure is integrated into the GA to accelerate convergence. The proposed GA has been tested on benchmark instances, and the computational results show that it gives better convergence than existing heuristics. 展开更多
关键词 Genetic algorithm (GA) pheromone-based crossover local search pickup and delivery traveling salesman problem(TSP).
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Finding the Minimum Ratio Traveling Salesman Tour by Artificial Ants 被引量:3
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作者 马良 崔雪丽 姚俭 《Journal of Systems Engineering and Electronics》 SCIE EI CSCD 2003年第3期24-27,共4页
Ants of artificial colony are able to generate good solutions to the famous traveling salesman problem (TSP). We propose an artificial ants algorithm for solving the minimum ratio TSP, which is more general than the s... Ants of artificial colony are able to generate good solutions to the famous traveling salesman problem (TSP). We propose an artificial ants algorithm for solving the minimum ratio TSP, which is more general than the standard TSP in combinatorial optimization area. In the minimum ratio TSP, another criterion concerning each edge is added, that is, the traveling salesman can have a benefit if he travels from one city to another. The objective is to minimize the ratio between total costs or distances and total benefits. The idea of this type of optimization is in some sense quite similar to that of traditional cost-benefit analysis in management science. Computational results substantiate the solution quality and efficiency of the algorithm. 展开更多
关键词 Minimum ratio TraveLing salesman problem Ant algorithm.
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Collision-free Scheduling of Multi-bridge Machining Systems: A Colored Traveling Salesman Problem-based Approach 被引量:2
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作者 Jun Li Xianghu Meng Xing Dai 《IEEE/CAA Journal of Automatica Sinica》 SCIE EI CSCD 2018年第1期139-147,共9页
Multi-bridge machining systems(MBMS) have gained wide applications in industry due to their high production capacity and efficiency. They contain multiple bridge machines working in parallel within their partially ove... Multi-bridge machining systems(MBMS) have gained wide applications in industry due to their high production capacity and efficiency. They contain multiple bridge machines working in parallel within their partially overlapping workspaces.Their scheduling problems can be abstracted into a serial-colored travelling salesman problem in which each salesman has some exclusive cities and some cities shared with its neighbor(s). To solve it, we develop a greedy algorithm that selects a neighboring city satisfying proximity. The algorithm allows a salesman to select randomly its shared cities and runs accordingly many times. It can thus be used to solve job scheduling problems for MBMS. Subsequently, a collision-free scheduling method is proposed to address both job scheduling and collision resolution issues of MBMS. It is an extension of the greedy algorithm by introducing time window constraints and a collision resolution mechanism. Thus, the augmented greedy algorithm can try its best to select stepwise a job for an individual machine such that no time overlaps exist between it and the job sequence of the neighboring machine dealt in the corresponding overlapping workspace; and remove such a time overlap only when it is inevitable. Finally, we conduct a case study of a large triplebridge waterjet cutting system by applying the proposed method. 展开更多
关键词 Collision resolution greedy algorithm modeling multiple traveling salesman problem SCHEDULING
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Traveling Salesman Problem Using an Enhanced Hybrid Swarm Optimization Algorithm 被引量:2
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作者 郑建国 伍大清 周亮 《Journal of Donghua University(English Edition)》 EI CAS 2014年第3期362-367,共6页
The traveling salesman problem( TSP) is a well-known combinatorial optimization problem as well as an NP-complete problem. A dynamic multi-swarm particle swarm optimization and ant colony optimization( DMPSO-ACO) was ... The traveling salesman problem( TSP) is a well-known combinatorial optimization problem as well as an NP-complete problem. A dynamic multi-swarm particle swarm optimization and ant colony optimization( DMPSO-ACO) was presented for TSP.The DMPSO-ACO combined the exploration capabilities of the dynamic multi-swarm particle swarm optimizer( DMPSO) and the stochastic exploitation of the ant colony optimization( ACO) for solving the traveling salesman problem. In the proposed hybrid algorithm,firstly,the dynamic swarms,rapidity of the PSO was used to obtain a series of sub-optimal solutions through certain iterative times for adjusting the initial allocation of pheromone in ACO. Secondly,the positive feedback and high accuracy of the ACO were employed to solving whole problem. Finally,to verify the effectiveness and efficiency of the proposed hybrid algorithm,various scale benchmark problems were tested to demonstrate the potential of the proposed DMPSO-ACO algorithm. The results show that DMPSO-ACO is better in the search precision,convergence property and has strong ability to escape from the local sub-optima when compared with several other peer algorithms. 展开更多
关键词 particle SWARM optimization(PSO) ant COLONY optimization(ACO) SWARM intelligence TRAVELING salesman problem(TSP) hybrid algorithm
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The Quantum Approximate Algorithm for Solving Traveling Salesman Problem 被引量:3
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作者 Yue Ruan Samuel Marsh +2 位作者 Xilin Xue Zhihao Liu Jingbo Wang 《Computers, Materials & Continua》 SCIE EI 2020年第6期1237-1247,共11页
The Quantum Approximate Optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization problems.It consists of interleaved unitary transformations induced by tw... The Quantum Approximate Optimization Algorithm(QAOA)is an algorithmic framework for finding approximate solutions to combinatorial optimization problems.It consists of interleaved unitary transformations induced by two operators labelled the mixing and problem Hamiltonians.To fit this framework,one needs to transform the original problem into a suitable form and embed it into these two Hamiltonians.In this paper,for the well-known NP-hard Traveling Salesman Problem(TSP),we encode its constraints into the mixing Hamiltonian rather than the conventional approach of adding penalty terms to the problem Hamiltonian.Moreover,we map edges(routes)connecting each pair of cities to qubits,which decreases the search space significantly in comparison to other approaches.As a result,our method can achieve a higher probability for the shortest round-trip route with only half the number of qubits consumed compared to IBM Q’s approach.We argue the formalization approach presented in this paper would lead to a generalized framework for finding,in the context of QAOA,high-quality approximate solutions to NP optimization problems. 展开更多
关键词 Quantum approximate optimization algorithm traveling salesman problem NP optimization problems
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Tabu Search Implementation on Traveling Salesman Problem and Its Variations: A Literature Survey 被引量:5
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作者 Sumanta Basu 《American Journal of Operations Research》 2012年第2期163-173,共11页
The Traveling Salesman Problem (TSP) and its allied problems like Vehicle Routing Problem (VRP) are one of the most widely studied problems in combinatorial optimization. It has long been known to be NP-hard and hence... The Traveling Salesman Problem (TSP) and its allied problems like Vehicle Routing Problem (VRP) are one of the most widely studied problems in combinatorial optimization. It has long been known to be NP-hard and hence research on developing algorithms for the TSP has focused on approximate methods in addition to exact methods. Tabu search is one of the most widely applied metaheuristic for solving the TSP. In this paper, we review the tabu search literature on the TSP and its variations, point out trends in it, and bring out some interesting research gaps in this literature. 展开更多
关键词 Tabu SEARCH TRAVELING salesman PROBLEM Vehicle ROUTING PROBLEM
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ISPO: A New Way to Solve Traveling Salesman Problem 被引量:3
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作者 Xiaohua Wang Aiqin Mu Shisong Zhu 《Intelligent Control and Automation》 2013年第2期122-125,共4页
This paper first introduces the concepts of mobile operators and mobile sequence, with which it redefines the rate of particle swarm optimization algorithm and the formula of position updating. Combining this discrete... This paper first introduces the concepts of mobile operators and mobile sequence, with which it redefines the rate of particle swarm optimization algorithm and the formula of position updating. Combining this discrete PSO algorithm with neighbors, the paper puts forward Hybrd Particle Swarm Optimization Algorithm, whose effectiveness is verified at the end of this paper. 展开更多
关键词 Mobile OPERATORS Particle SWARM Optimization TRAVELING salesman Problem
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Fast Algorithm for the Travelling Salesman Problem and the Proof of P = NP 被引量:1
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作者 Jinliang Wang 《Applied Mathematics》 2018年第12期1351-1359,共9页
In the theory of computational complexity, the travelling salesman problem is a typical one in the NP class. With the aid of a brand-new approach named “maximum-deleting method”, a fast algorithm is constructed for ... In the theory of computational complexity, the travelling salesman problem is a typical one in the NP class. With the aid of a brand-new approach named “maximum-deleting method”, a fast algorithm is constructed for it with a polynomial time of biquadrate, which greatly reduces the computational complexity. Since this problem is also NP-complete, as a corollary, P = NP is proved to be true. It indicates the crack of the well-known open problem named “P versus NP”. 展开更多
关键词 TRAVELLING salesman PROBLEM P versus NP PROBLEM NP-COMPLETE Computational Complexity Maximum-Deleting Method
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